Abstract
This paper investigates the application of motion planning techniques to the untangling of mathematical knots. Knot untangling can be viewed as a high-dimensional planning problem in reparameterizable configuration spaces. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths. We develop a probabilistic planner that is capable of untangling knots described by over four hundred variables and known difficult benchmarks in this area more quickly than has been achieved with minimization in the literature. The use of motion planning techniques was critical for the untangling. Our planner defines local goals and makes combined use of energy minimization and randomized tree-based planning. We also show how to produce candidates with minimal number of segments for a given knot. Finally, we discuss some possible applications of our untangling planner in computational topology, in the study of DNA rings and protein folding and for planning with flexible robots.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. Agol, J. Hass, and W. Thurston. The computational complexity of knot genus and spanning area, (preprint.).
C. N. Anerizis. The Mystery of Knots - Computer Programming for Knot Tabulation. Series on Knots and Everything. World Scientifical Publishing Co. Pte. Ltd., 1999.
M. Apaydin, D. Brutlag, C. Guestrin, D. Hsu, and J. Latombe. Stochastic roadmap simulation: An efficient representation and algorithm for analyzing molecular motion. In International Conference on Computational Molecular Biology (RECOMB), April 2002.
M. Apaydin, A. Singh, D. Brutlag, and J. Latombe. Capturing molecular energy landscapes with probabilistic conformal roadmaps. In IEEE International Conference on Robotics and Automation (ICRA), May 2001.
O. Bayazit, J.-M. Lien, and N. Amato. Probabilistic roadmap motion planning for deformable objects. In IEEE International Conference on Robotics and Automation, 2002.
M. Bern, D. Eppstein, and al. Emerging challenges in computational topology, 1999.
J. S. Birman, P. Boldi, M. Rampichini, and S. Vigna. Towards an implementation of the b-h algorithm for recognizing the unknot. In KNOTS-2000, 2000.
Z. Butler, K. Kotay, D. Rus, and K. Tomita. Cellular automata for decentralized control of self-reconfigurable robots. In IEEE International Conference on Robotics and Automation, 2001.
X. Dai and Y. Diao. The minimum of knot energy functions. Journal of Knot Theory and its Ramifications, 9 (6): 713–724, 2000.
R. Deibler, S. Rahmati, and E. Zechiedrich. Topoisomerase iv, alone, unknots dna in escherichia coli. Genes and Development, 15: 748–761, 2001.
Y. Diao, C. Ernst, and J. Rensburg. In search of a good polygonal knot energy. Journal of Knot Theory and its Ramifications, 6(5):633–657, 1997.
R. Grzeszczuk, M. Huang, and L. Kauffman. Untangling knots by stochastic energy optimization. IEEE Visualization, pages 279–286, 1996.
R. Grzeszczuk, M. Huang, and L. Kauffman. Physically-based stochastic simplification of mathematical knots. IEEE Transactions on Visualization and Computer Graphics, 3 (3): 262–278, 1997.
J. Hass and J. Lagarias. The number of reidemeister moves needed for unknotting. (preprint.).
J. Hass, J. C. Lagarias, and N. Pippenger. The computational complexity of knot and link problems. In IEEE Symposium on Foundations of Computer Science, pages 172–181, 1997.
J. Hoste and M. Thistlethwaite. The first 1,701,936 knots. Math. Intelligencer, 20 (4): 33–48, 1998.
D. Hsu. Randomized Single-Query Motion Planning In Expansive Spaces. PhD thesis, Department of Computer Science, Stanford University, 2000.
D. Hsu, R. Kindel, J. Latombe, and S. Rock. Control-based randomized motion planning for dynamic environments. In Algorithmic and Computational Robotics: New Directions: The Fourth International Workshop on the Algorithmic Foundations of Robotics, pages 247–264, 2001.
D. Hsu, J. Latombe, and R. Motwani. Path planning in expansive spaces. In Proc. IEEE Int’l Conf. on Robotics and Automation, pages 2719–2726, 1997.
F. Jaeger, D. L. Vertigan, and D. Welsh. On the computational complexity of the jones and tutte polynomials. In Math. Proc. Camb. Phil. Soc108, pages 35–53, 1990.
R. Jenkins. A dynamic approach to calculating the homfly polynomial for directed knots and links. Master’s thesis, Carnegie Mellon University, 1989.
L. E. Kavraki, P. Svestka, J.-C. Latombe, and M. H. Overmars. Probabilistic roadmaps for path planning in high-dimensional configuration spaces. Transaction on Robotics and Automation, 12 (4): 566–580, June 1996.
J. J. Kuffner and S. M. LaValle. Randomized kinodynamic planning. In Proc. IEEE Int’l Conf. on Robotics and Automation, pages 473–479, 1999.
J. J. Kuffner and S. M. LaValle. RRT-connect: An efficient approach to single-query path planning. In Proc. IEEE Int’l Conf. on Robotics and Automation, 2000.
J. J. Kuffner and S. M. LaValle. Randomized kinodynamic planning. International Journal of Robotics Research, 20 (5): 378–400, May 2001.
J. J. Kuffner and S. M. LaValle. Rapidly exploring random trees: Progress and prospects. In Algorithmic and Computational Robotics: New Directions: The Fourth International Workshop on the Algorithmic Foundations of Robotics, pages 293–308, 2001.
A. Ladd and L. Kavraki. A measure theoretic analysis of prm. In IEEE International Conference on Robotics and Automation, May 2002.
F. Lamiraux and L. Kavraki. Planning paths for elastic objects. International Journal of Robotics Research, 20 (3), 2001.
W. Lickorish. An Introduction to Knot Theory. Springer, 1997.
T. Ligocki and J. A. Sethian. Recognizing knots using simulated annealing. Journal of Knot Theory and Its Ramifications, 3 (4): 477–495, 1994.
J. Phillips, A. Ladd, and L. Kavraki. Simulated knot tying. In IEEE International Conference on Robotics and Automation, May 2002.
L. Postow, B. Peter, and N. Cozzarelli. Knot what we thought before: The twisted story of replication. BioEssays, 21: 805–808, 1999.
G. Sanchez and J.-C. Latombe. A single-query bi-directional probabilistic roadmap planner with lazy collision checking. In ISRR, 2001.
R. Scharein. Interactive Topological Drawing. PhD thesis, University of British Columbia, 1988.
J. Simon. Energy functions for polygonal knots. J. Knot Theory and its Ramif., 3: 299–320, 1994.
G. Song and N. Amato. Using motion planning to study protein folding pathways. In International Conference on Computational Molecular Biology (RE- COMB), pages 287–296, April 2001.
M. Teodoro, G. Phillips, and L. Kavraki. A dimensionality reduction approach to modeling protein flexibility. In International Conference on Computational Molecular Biology (RECOMB), April 2002.
S. Vassilvitskii, J. Suh, and M. Yim. A complete, local and parallel reconfiguration algorithm for cube style modular robots. In IEEE International Conference on Robotics and Automation (ICRA), May 2002.
J. Walter, B. Tsai, and N. Amato. Choosing good paths for fast distributed reconfiguration of hexagonal met amorphic robots. In IEEE International Conference on Robotics and Automation, 2002.
Y.-Q. Wu. Ming user manual, www.math.uiowa.edu/wu/ming/ming.pdf, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Ladd, A.M., Kavraki, L.E. (2004). Motion Planning for Knot Untangling. In: Boissonnat, JD., Burdick, J., Goldberg, K., Hutchinson, S. (eds) Algorithmic Foundations of Robotics V. Springer Tracts in Advanced Robotics, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45058-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-45058-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07341-0
Online ISBN: 978-3-540-45058-0
eBook Packages: Springer Book Archive